A Lie group representation where trace cannot send non-conjugate elements to the same thing

Question

Let $G$ be a Lie group and $g,h\in G$. Is there a representation (finite dimensional, over $\mathbb R$ is preferable but any other field (like $\mathbb C$) or if necessary, infinite dimensional representations will do) $\rho\colon G\to\mathrm{Aut}\left(V\right)$ such that \begin{equation} \mathrm{Tr}\left(\rho\left(g\right)\right)=\mathrm{Tr}\left(\rho\left(h\right)\right)\Longleftrightarrow\exists k\text{ such that }g=khk^{-1} \end{equation}